Linear Algebra Done Right: Notation 1.23

The question has already been answered, but here is a little context: the notation $Y^X$ is a little odd to denote the set of functions $X \rightarrow Y$. The origin of this notation lies in combinatorics, and more generally set theory.

Let $[n]$ be the set $\{1,\dots, n\}$. It is a classic motivational exercise in combinatorics to count the size of the set $\{f | f: [n] \rightarrow [m]\}$, and it turns out the answer is $m^n$ many functions (can you prove this?). So when $X$ and $Y$ are finite, we have the wonderful correspondence:

$$\left| Y^X\right| = |Y|^{|X|}$$

Moreover, once you have some cardinal arithmetic under your belt, we may extend this correspondence to infinite sets as well.

It means that if $f\in \mathbb{F} ^S$ then $f$ is a function from the set $S$ to a field (think $\mathbb{R}$ or $\mathbb{C}$ for example), $f:S\to\mathbb{F} $. Also, $\mathbb{F} ^S$, with the sum (sum of functions) and product by scalar (scalar times function) as defined in 1.23 is a vector space.